Find the exact solution of the equation in its simplest form: 3^x * e^4x = e^7.
First take natural logs of both sides, giving us: ln(3x * e4x) = ln(e7). The RHS can be simplified using the definition of natural logs, so ln(e7) = 7.
We can then apply the product rule for logs (ln(a * b) = ln(a) + ln(b)), which gives: ln(3x) + ln(e4x) = 7.
Using the power rule, (ln(ab) = b*ln(a)) and the def. of natural logs, the equation can be simplified further: xln(3) + 4x = 7.
Factorise by taking out the factors of x to give: x(ln(3) + 4) = 7.
Then divide both sides by (ln(3) + 4) to get the equation with x as the subject: x = 7 / (ln(3) + 4).
We now have an exact value for x and we can check the answer by substituting it back into the original equation and checking that we get e7.
HM
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